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Chapter 2: Problem 7

Find real numbers \(a\) and \(b\) such that the equation is true. $$a+b i=-12+7 i$$

Short Answer

Expert verified
From the comparison, \(a = -12\) and \(b = 7\). Therefore, the real numbers \(a\) and \(b\) that satisfy the equation are \(-12\) and \(7\), respectively.

Step by step solution

01

Compare the real parts

The real part of the complex number on the left side is \(a\) and on the right side is \(-12\). Since the real parts on both sides need to be equal, we get the equation: \(a = -12\).
02

Compare the imaginary parts

The imaginary part of the complex number on the left side is \(b\) and on the right side is \(7\). Since the imaginary parts on both sides need to be equal, we get the equation: \(b = 7\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Real Part
When working with complex numbers, every complex number is composed of two essential components: the real part and the imaginary part. The real part is the component that does not involve the imaginary unit \(i\), which is defined by \(i^2 = -1\). In the complex number \(a + bi\), \(a\) represents the real part.
  • In practical scenarios, if you have a complex number like \(-12 + 7i\), the real part is simply \(-12\).
To determine equality for complex numbers, the real parts on either side of the equation must be equal. This means that if we have two complex numbers like \(a + bi = -12 + 7i\), for them to be equal, \(a\) (the real part) must be equal to \(-12\). Hence, from this condition, we establish that \(a = -12\).
Imaginary Part
The imaginary part of a complex number involves the imaginary unit \(i\). In \(a + bi\), \(b\) represents the imaginary part. Though described as "imaginary," this part is crucial in many applications, including engineering and physics.
  • For the complex number \(-12 + 7i\), the imaginary part is \(7\).
The imaginary part dictates the component of the complex number that involves the square root of negative one. Just like the real part, for two complex numbers to be equal, their imaginary parts must be identical. Therefore, considering the comparison \(a + bi = -12 + 7i\), the imaginary part \(b\) must equal \(7\). Thus, \(b = 7\).
Equality of Complex Numbers
Understanding the equality of complex numbers is pivotal when solving equations involving them. Two complex numbers, say \(a + bi\) and \(c + di\), are deemed equal if both their real and imaginary parts are identical.
  • This boils down to the conditions \(a = c\) and \(b = d\).
Referring back to our example, the equation \(a + bi = -12 + 7i\) requires that the real parts \(a\) and \(-12\) are the same, as well as the imaginary parts \(b\) and \(7\). This breakdown results from the fundamental definition of complex number equality and is used to solve such comparisons, determining that \(a = -12\) and \(b = 7\). Recognizing these simple rules can immensely aid in tackling more complex problems involving complex numbers.

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